Some Second Order Set Theory

Set theory is the study of sets, particularly the transfinite, with a focus on well-founded transfinite recursion. Began with Cantor in late 19th century, matured in mid 20th century. Set theory today is vast: independence, large cardinals, forcing, combinatorics, the continuum, descriptive set theory,... Set theory also serves as an ontological foundation for all (or much of) mathematics. Mathematical objects can be viewed as having a set theoretical essence. Natural numbers, rationals, reals, functions, topological spaces, etc. Mathematical precision often means specifying an object in set theory. Set theory consequently speaks to or with other mathematical subjects, particularly on foundational matters. Introduction Modal Logic of Forcing Set-Theoretic Geology Models of set theory The fundamental axioms of set theory are the Zermelo-Fraenkel ZFC axioms, which concern set existence. Each model of ZFC is an entire mathematical world, in which any mathematician could be at home. A mathematical statement ϕ can be proved independent of ZFC by providing a model of ZFC in which ϕ holds and another in which ϕ fails. For example, Gödel provided a model of ZFC in which the Continuum Hypothesis holds, and Cohen provided one in which it fails. Set theorists have powerful methods to construct such models. e.g. forcing (Cohen 1963). We now have thousands. As set theory has matured, the fundamental object of study has become: the model of set theory. Second order set theory Set theory now exhibits a category-theoretic nature. What we have is a vast cosmos of models of set theory, each its own mathematical universe, connected by forcing extensions and large cardinal embeddings. The thesis of this talk is that, as a result, set theory now exhibits an essential second-order nature. Two emerging developments Two emerging developments are focused on second-order features of the set theoretic universe. Modal Logic of forcing. Upward-oriented, looking from a model of set theory to its forcing extensions. Set-theoretic geology. Downward-oriented, looking from a model of set theory down to its ground models. This analysis engages pleasantly with various philosophical views on the nature of mathematical existence. In particular, the two perspectives are unified by and find motivation in a multiverse view of set theory, the philosophical view that there are many set-theoretic worlds.